By using Holley Stroock perturbation theorem [1]},
we have \(U(x)\)  is log-Sobolev on \(\mathbb {R}^{d}\)  with constant
\(\gamma _{3}=\frac{2\gamma e^{-\left(2\sum _{i}L_{i}R^{1+\alpha _{i}}+4L_{N}R^{2}+4LR^{1+\alpha }\right)}}{[A\gamma +(B+2)e^{4\left(2\sum _{i}L_{i}R^{1+\alpha _{i}}+4L_{N}R^{2}+4LR^{1+\alpha }\right)})]}.\) 
